qt,t+1(xt, xt+1) we applied a damping of = 0.9 for the temporal messages αt and βt to improve convergence. Generally we experience fast convergence for all expectation prop-agation algorithms for the various models (e.g., qX and qAZ). It seems that the (known) slow convergence of the variational Bayes approach dominates the overall convergence when updating the forms of qX, qAZ, qB and qQ.
D Scheduling options
Figure 15 illustrates the approximate marginals in QX and the messages used in the cor-responding message-passing algorithm. The scheduling options from the main text can be summarised as follows.
Dynamic scheduling
(i) pick the message with the largest last update in absolute value from all αt+1, βt, λlt+1,j and λ0t+1,j
(ii) choose from the following steps – if λ0t+1,j is chosen
- compute ˜qt,t+1 and update λlt+1,j, αt+1 and βt - compute ˜qt+1,j and update λ0t+1,j
– if λlt+1,j is chosen,
- compute ˜qt+1,j and update λ0t+1,j
- compute ˜qt,t+1 and update λlt+1,j, αt+1 and βt – if αt+1 is chosen
- compute ˜qt+1,t+2 and update λlt+2,j, αt+2 and βt+1
- compute ˜qt,t+1 and update λlt+1,j, βt and αt+1
– if βt is chosen
- compute ˜qt−1,t and update λlt,j, βt−1 and αt
- update ˜qt,t+1 and update λlt+1,j, αt+1 and βt
(iii) repeat (i) and (ii) until the largest update in step (i) is below a threshold
Sequential scheduling
(i) run until convergence: compute ˜qt,t+1and update λlt+1,j for all j ∈ {1, . . . , n}, compute
˜
qt+1,j and update λ0t+1,j
(ii) update the αt+1 or βt message depending on whether doing a forward or a backward step
(iii) repeat (i) and (ii) in a forward-backward fashion till the absolute or relative update in all αt+1, βt, λlt+1,j and λ0t+1,j is below a threshold
Note that after the initial forward sweep, the iteration in step (i) converges in a few steps.
Static Scheduling
(i) run till convergence in a forward-backward fashion: compute ˜qt,t+1 and update αt+1, βt
and λlt+1,j; updating backward messages βt in a forward iteration is not necessary and likewise for the forward messages αt+1 in the backward iteration
(ii) for all j ∈ {1, . . . , n}, t ∈ {1, . . . , T } update ˜qt,j and λ0t,j
(iii) repeat (i) and (iii) till the absolute or relative update in all αt+1, βt, λlt+1,j and λ0t+1,j is below a threshold
References
J. Ahn, T. D. Johnson, D. Bhavnani, J. N. S. Eisenberg, and B. Mukherjee. A space-time point process model for analyzing and predicting case patterns of diarrheal disease in northwestern Ecuador. Spatial and Spatio-temporal Epidemiology, 9(0):23 – 35, 2014.
P. R. Amestoy, T. A. Davis, and I. S. Duff. An approximate minimum degree ordering algorithm. SIAM Journal of Matrix Analysis and Applications, 17(4):886–905, 1996.
C. Andrieu, A. Doucet, and R. Holenstein. Particle Markov chain Monte Carlo methods.
Journal of the Royal Statistical Society (Series B), 72(3), 2010.
A. Armagan, M Clyde, and D. B. Dunson. Generalized Beta mixtures of Gaussians. In Advances in Neural Information Processing Systems 24, pages 523–531. Curran Associates, Inc., New York, 2011.
P. Baudains, S. D. Johnson, and A. M. Braithwaite. Geographic patterns of diffusion in the 2011 London riots. Applied Geography, 45:211–219, 2013.
I. Brainman and S. Toledo. Nested-dissection orderings for sparse LU with partial pivoting.
SIAM Journal on Matrix Analysis and Applications, 23(4):998–1012, 2002.
D. Bueche, N. Sukumar, and B. Moran. Dispersive properties of the natural element method.
Computational Mechanics, 25(2):207–219, 2000.
B. Cseke and T. Heskes. Approximate marginals in latent Gaussian models. Journal of Machine Learning Research, 12:417–454, 2011.
E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In Pro-ceedings of the 1969 24th National Conference, pages 157–172. ACM, 1969.
J. Dahl, L. Vandenberghe, and V. Roychowdhury. Covariance selection for non-chordal graphs via chordal embedding. Optimization Methods Software, 23(4):501–520, 2008.
T. A. Davis. Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2).
Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006.
P. Diggle, B. Rowlingson, and T. Su. Point process methodology for on-line spatio-temporal disease surveillance. Environmetrics, 16(5):423–434, 2005.
A. George. Nested dissection of a regular finite element mesh. SIAM Journal on Numerical Analysis, 10(2):345–363, 1973.
M. Gerven, B. Cseke, R. Oostenveld, and T. Heskes. Bayesian source localization with the multivariate Laplace prior. In Advances in Neural Information Processing Systems 22, pages 1901–1909, 2009.
M. Gerven, A. Bahramisharif, J. Farquhar, and T. Heskes. Donders Machine Learning Toolbox, 2012. URLhttps://github.com/distrep/DMLT. Last accessed 1 June 2015.
M. Girolami and B. Calderhead. Riemann manifold Langevin and Hamiltonian Monte Carlo methods. Journal of the Royal Statistical Society (Series B), 73(2):123–214, 2011.
J. Hartikainen, J. Riihim¨aki, and S. S¨arkk¨a. Sparse spatio-temporal Gaussian processes with general likelihoods. In Proceedings of the 21th International Conference on Artificial Neural Networks, pages 193–200, 2011.
D. Hern´andez-Lobato, J. M. Hern´andez-Lobato, and P. Dupont. Generalized spike-and-slab priors for bayesian group feature selection using expectation propagation. Journal of Machine Learing Research, 14(1):1891–1945, 2013.
T. Heskes. Stable fixed points of loopy belief propagation are minima of the Bethe free energy.
In Advances in Neural Information Processing Systems 15, pages 359–366, Cambridge, MA, 2003. The MIT Press.
T. Heskes. On the uniqueness of loopy belief propagation fixed points. Neural Computation, 16:2379–2413, 2004.
T. Heskes. Convexity arguments for efficient minimization of the Bethe and Kikuchi free energies. Journal of Artificial Intelligence Research, 26(1), June 2006.
T. Heskes and O. Zoeter. Expectation propagation for approximate inference in dynamic Bayesian networks. In Proceedings of the 18th Conference in Uncertainty in Artificial Intelligence, pages 216–223, 2002.
T. Heskes, M. Opper, W. Wiegerinck, O. Winther, and O. Zoeter. Approximate inference techniques with expectation constraints. Journal of Statistical Mechanics: Theory and Experiment, 2005.
M. B. Hooten and C. K. Wikle. A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove. Envi-ronmental and Ecological Statistics, 15(1):59–70, 2008.
M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction to variational methods for graphical models. Machine Learning, 37(2):183–233, 1999.
D. Koller and N. Friedman. Probabilistic Graphical Models: Principles and Techniques. MIT Press, 2009.
F. R. Kschischang, B. J. Frey, and H. A. Loeliger. Factor graphs and the sum-product algorithm. IEEE Transactions on Information Theory, 47(2), 2001.
M. Kuss and C. E. Rasmussen. Assessing approximate inference for binary Gaussian process classification. Journal of Machine Learning Research, 6:1679–1704, 2005.
M. Kuss, T. Pfingsten, L. Csato, and C. E. Rasmussen. Approximate inference for robust Gaussian process regression. Technical Report 136, Max Planck Institute for Biological Cybernetics, 2005.
S. L. Lauritzen. Graphical Models. Oxford University Press, Oxford, UK, 1996.
F. Lindgren, H. Rue, and J. Lindstr¨om. An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. Journal of the Royal Statistical Society (Series B), 73(4):423–498, 2011.
T. P. Minka. A Family of Algorithms for Approximate Bayesian Inference. PhD thesis, MIT, MA, 2001.
T. P. Minka. Power EP. Technical report, Microsoft Research Ltd., Cambridge, UK, MSR-TR-2004-149, October 2004.
T. P. Minka. Divergence measures and message passing. Technical Report MSR-TR-2005-173, Microsoft Research Ltd., Cambridge, UK, 2005.
J. Møller, A. R. Syversveen, and R. P. Waagepetersen. Log-Gaussian Cox processes. Scan-dinavian Journal of Statistics, 25(3):451–482, 1998.
K. Murphy, Y. Weiss, and M. I. Jordan. Loopy belief propagation for approximate inference:
An empirical study. In Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence, volume 9, pages 467–475, 1999.
K. P. Murphy and Y. Weiss. The factored frontier algorithm for approximate inference in DBNs. In Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence, pages 378–385, 2001.
M. Opper and C. Archambeau. The variational Gaussian approximation revisited. Neural Computation, 21(3):786–792, 2009.
M. Opper and O. Winther. Gaussian processes for classification: Mean-field algorithms.
Neural Computation, 12(11):2655–2684, 2000.
M. Opper and O. Winther. Expectation consistent approximate inference. Journal of Ma-chine Learning Research, 6:2177–2204, 2005.
C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press, Cambridge, MA, 2005.
H. Rue and L. Held. Gaussian Markov Random Fields Theory and Applications. Chapman and Hall/CRC Press, 2005.
H. Rue, S. Martino, and N. Chopin. Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. Journal of The Royal Statis-tical Society (Series B), 71(2):319–392, 2009.
L. K. Saul, T. Jaakkola, and M. I. Jordan. Mean field theory for sigmoid belief networks.
Journal of Artificial Intelligence Research, 4(1), 1996.
S. Schutte and N. B. Weidmann. Diffusion patterns of violence in civil wars. Political Geography, 30:143–152, 2011.
M. W. Seeger. Bayesian inference and optimal design for the sparse linear model. Journal of Machine Learning Research, 9:759–813, 2008.
D. Simpson, J. Illian, F. Lindgren, S. Sørbye, and H. Rue. Going off grid: Computationally efficient inference for log-Gaussian Cox processes. arXiv:1111.0641, 2011.
D. Simpson, I. W. Turner, C. M. Strickland, and A. N. Pettitt. Scalable iterative methods for sampling from massive Gaussian random vectors. arXiv:1312.1476, 2013.
K. Takahashi, J. Fagan, and M.-S. Chin. Formation of a sparse impedance matrix and its application to short circuit study. In Proceedings of the 8th PICA Conference, 1973.
M. Wainwright, T. Jaakkola, and A. Willsky. Tree-reweighted belief propagation algorithms and approximate ML estimation via pseudo-moment matching. In C. Bishop and B. Frey, editors, Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics. Society for Artificial Intelligence and Statistics, 2003.
W. Wiegerinck and T. Heskes. Fractional belief propagation. In S. Becker, S. Thrun, and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, pages 438–445, Cambridge, MA, 2003. The MIT Press.
C. K. Wikle. A kernel-based spectral model for non-Gaussian spatio-temporal processes.
Statistical Modelling, 2(1):299–314, 2002.
C. K. Wikle, R. F. Milliff, D. Nychka, and L. M. Berliner. Spatiotemporal hierarchical Bayesian modeling tropical ocean surface winds. Journal of the American Statistical As-sociation, 96(454):382–397, 2001.
T. Wist and H. Rue. Specifying a Gaussian Markov random field by a sparse Cholesky tri-angle. Communications in Statistics: Simulation and Computation, 35(1):161–176, 2006.
J. S. Yedidia, W. T. Freeman, and Y. Weiss. Generalized belief propagation. In Advances in Neural Information Processing Systems 12, pages 689–695. The MIT Press, 2000.
A. Ypma and T. Heskes. Novel approximations for inference in nonlinear dynamical systems using expectation propagation. Neurocomputing, 69(1):85–99, 2005.
K. Yuan, M. Girolami, and M. Niranjan. Markov chain Monte Carlo methods for state-space models with point process observations. Neural Computation, 24(6):1462–1486, 2012.
A. Zammit-Mangion, M. Dewar, V. Kadirkamanathan, and G. Sanguinetti. Point process modelling of the Afghan war diary. Proceeding of the National Academy of Sciences, 109 (31):12414–12419, 2012a.
A. Zammit-Mangion, G. Sanguinetti, and V. Kadirkamanathan. Variational estimation in spatiotemporal systems from continuous and point-process observations. IEEE Transac-tions on Signal Processing, 60(7):3449–3459, 2012b.